Sunday, April 23, 2017

[Alliance:HotS] Theorycrafting Critical DoTs

I've been playing a lot of Alliance: Heroes of the Spire the past couple of months. It's a nifty Summoners War clone made by a North American company, which has been pretty cool because we get to talk to the devs relatively often--as a dev myself, I'm pretty appreciative of that access.So, you summon heroes, level them up, gear them up with up to 6 pieces of gear, and let them loose on other teams and dungeons. The thing about gear is you can wear x number of a type of gear to get a benefit. For example, Bone gear will increase your hero's Health by an extra 20% for every 2 pieces of Bone gear you have on. Some more powerful sets require 4 pieces, making them mutually exclusive with other 4 piece sets--such as Swiftsteel (25% chance of performing an extra ability 1 attack every time you attack), or Titanguard (Transfer 30% of damage done to your party to you, reduce incoming damage by 15%).

However, one of the more interesting mechanics I've found in the game is that using a specific item set--Witchstone--your buffs/debuffs can critical strike, making them either 40% ~ 66%ish more powerful depending on the buff/debuff, or undispellable if it's not a numeric buff.A prime example of this are Damage Over Time debuffs, aka DoTs. The regular DoT always does 5% of the target's max health in damage each round, but a critical DoT does 7% per round (an increase of 40% total damage). DoTs are great at shredding most PvE bosses, as they tend to be a big target with lots of Armor (damage reduction), and DoTs ignore Armor.

Razormane, Flameclaw, and Icefang are plentiful and, against the right targets, powerful.

As such, lots of people like to use Razormane or Flameclaw to take out bosses. The benefit of these cats is that they're quite common--they're available from the worst pull you can make, and even drop from some dungeons, and their main attack has a 30% chance to apply a DoT (and can apply a second DoT if they were stealthed when they attacked).

Where this becomes really interesting is Swiftsteel (25% chance of an extra attack) vs. Witchstone (If your DoT crits, it'll be 7% DoT). Are the extra attacks better than the critical DoT? Let's do some math.Making this all slightly more complex is the fact that debuffs may not always land. The to-hit of a debuff is Base Chance + (Aim - Enemy Block). Pretty close to everything has a base 15% to block (Bosses actually hit 25% at the highest floor you can fight them at).So we have two possible stats for our DoT to scale from: Aim (Hit%), and Crit%. To make this easier, we'll ignore stealth for Flameclaw, and just assume spamming the first ability over and over.Varying over Crit%Let's assume we have a 100% chance to apply a DoT (125% Aim against a Floor 6 boss), and vary the critical strike percentage.For Witchstone, we have an Crit%, aka x, chance to apply a 7% DoT, otherwise it's a 5% DoT:

For Swiftsteel, it never crits, so the amount of damage we can apply is based on the Swiftsteel proc rate:

Now, there's a small flaw in this math that I'm going to glaze over, which is that these expected health percentage damage values are an average over a lot of samples. In a single fight where you might get 5 - 10 turns, the variation is going to be much higher, so take this with a grain of salt. But over a lot of fights, we can work without dealing with that flaw.In any case, since Witchstone's damage is varying over Crit%, but Swiftsteel doesn't, that suggests there should be a solid inflection point where Witchstone will generally outperform Swiftsteel for DoT damage:

So, assuming we'll always apply a DoT, Witchstone will usually outperform Swiftsteel once you reach 62.5% chance to critical strike your DoT.Varying over Hit%Let's assume we have a 100% chance to critical strike, and vary over Hit%, aka y. I'll ignore the Block/Aim/Base Chance portion, and work with the Hit% directly to make life a little easier.For Witchstone, this is simply, since it always crits:

For Swiftsteel, this means we scaled both the regular attack and the 25% proc attack by Hit%:

Comparing the two, we actually find out that Witchstone simply scales faster than Swiftsteel with Hit%. The only value of y where an inflection point can exist is y = 0:

We'll see this fact crop up again when we try to vary over both Crit% and Hit%.

Varying over both Crit% and Hit%Remember, x is Crit%, y is Hit%Witchstone:

Swiftsteel:

Equating the two:

Almost immediately we notice we can divide the entire equation by y, removing the variable. Basically, Hit% is meaningless to how they scale relatively to each other. Which means that 62.5% Crit% is the magic number where the two become equivalent for a really basic scenario.How Reality Completely Breaks My ModelOf course, Swiftsteel is more interesting than I've allowed for in my modeling. It can actually proc off any attack, meaning that if you use Prowl, you could end up making an extra attack, where the Witchstone build might not get anything except an undispellable Stealth buff. But on the other hand, if you're using Swiftsteel and it procs on a move that has no target, I believe it picks a target at random, so it might be a wash depending on who it targets (unless you only have one target).This actually makes Swiftsteel significantly more valuable than that 62.5% Crit% inflection point would have you believe, as a fully skilled up cat on auto-battle will only use it's A1 ability every other round based on cooldown rotation, which if you squint kinda makes it like a 50%ish proc rate instead of 25% proc rate--if we count each A1 usage as a double chance to proc instead, which is a small fallacy but close enough for demonstrable purposes--which would actually make the inflection point 130% Crit%, which is absurd as anything above 100% is wasted (also, good luck hitting that much Crit%). It also doesn't take into account the extra initial damage that each Swiftsteel attack would grant as well, though in the case of the cats, it's usually small enough to be negligible.But Witchstone has other benefits. For example, Sunslash, the Order cat, has an A3 that Marks all targets for 3 rounds, increasing the amount of damage anybody does to that target by 30%; 50% on a critical strike if you have Witchstone, which means a significant chunk of extra damage overall to potentially the entire enemy party, which makes Witchstone a better bet for Sunslash for overall DPS (assuming you can stick those debuffs). He'll likely have fewer DoTs, but critical DoTs will help make up that difference a little.However, the 62.5% Crit% inflection is something to remember if we run into other DoT classes. Enough Crit%, and Witchstone will outstrip Swiftsteel's performance. But at the end of the day, it also comes down to what other abilities your hero is rocking, and what you need that hero for.But if you're just using Flameclaw/Razormane for Boss Shredding DoT application, Swiftsteel is the way to go. #Theorycraft, #AllianceHotS